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TensorComplexes :: flattenedGenericTensor

flattenedGenericTensor -- Make a generic tensor of given format

Synopsis

Description

Given a list L = {a, b1,…, bn} of positive integers with a= sumi bi, and a field (or ring of integers) kk, the script creates a polynomial ring S over kk with a×b1×…×bn variables, and a generic map

f: A →B1⊗…⊗Bn

of LabeledModules over S, where A is a free LabeledModule of rank a and Bi is a free LabeledModule of rank bi. We think of f as representing a tensor of type (a,b1,…,bn) made from the elementary symmetric functions.

The format of F is the one required by tensorComplex1, namely f: A →B1⊗…⊗Bn, with a = rank A, bi = rank Bi.
i1 : kk = ZZ/101

o1 = kk

o1 : QuotientRing
i2 : f = flattenedGenericTensor({5,2,1,2},kk)

o2 = | x_(0,0,0,0) x_(1,0,0,0) x_(2,0,0,0) x_(3,0,0,0) x_(4,0,0,0) |
     | x_(0,0,0,1) x_(1,0,0,1) x_(2,0,0,1) x_(3,0,0,1) x_(4,0,0,1) |
     | x_(0,1,0,0) x_(1,1,0,0) x_(2,1,0,0) x_(3,1,0,0) x_(4,1,0,0) |
     | x_(0,1,0,1) x_(1,1,0,1) x_(2,1,0,1) x_(3,1,0,1) x_(4,1,0,1) |

                                                                                                                                                                                                                        4                                                                                                                                                                                                                  5
o2 : Matrix (kk[x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       ])  <--- (kk[x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       ])
                 0,0,0,0   0,0,0,1   0,1,0,0   0,1,0,1   1,0,0,0   1,0,0,1   1,1,0,0   1,1,0,1   2,0,0,0   2,0,0,1   2,1,0,0   2,1,0,1   3,0,0,0   3,0,0,1   3,1,0,0   3,1,0,1   4,0,0,0   4,0,0,1   4,1,0,0   4,1,0,1              0,0,0,0   0,0,0,1   0,1,0,0   0,1,0,1   1,0,0,0   1,0,0,1   1,1,0,0   1,1,0,1   2,0,0,0   2,0,0,1   2,1,0,0   2,1,0,1   3,0,0,0   3,0,0,1   3,1,0,0   3,1,0,1   4,0,0,0   4,0,0,1   4,1,0,0   4,1,0,1
i3 : numgens ring f

o3 = 20
i4 : betti matrix f

            0 1
o4 = total: 4 5
        -1: . 5
         0: 4 .

o4 : BettiTally
i5 : S = ring f

o5 = S

o5 : PolynomialRing
i6 : tensorComplex1 f

o6 = | -x_(0,1,0,0)x_(1,0,0,0)+x_(0,0,0,0)x_(1,1,0,0)                                               0                                                                                            -x_(0,1,0,0)x_(2,0,0,0)+x_(0,0,0,0)x_(2,1,0,0)                                               0                                                                                            -x_(1,1,0,0)x_(2,0,0,0)+x_(1,0,0,0)x_(2,1,0,0)                                               0                                                                                            -x_(0,1,0,0)x_(3,0,0,0)+x_(0,0,0,0)x_(3,1,0,0)                                               0                                                                                            -x_(1,1,0,0)x_(3,0,0,0)+x_(1,0,0,0)x_(3,1,0,0)                                               0                                                                                            -x_(2,1,0,0)x_(3,0,0,0)+x_(2,0,0,0)x_(3,1,0,0)                                               0                                                                                            -x_(0,1,0,0)x_(4,0,0,0)+x_(0,0,0,0)x_(4,1,0,0)                                               0                                                                                            -x_(1,1,0,0)x_(4,0,0,0)+x_(1,0,0,0)x_(4,1,0,0)                                               0                                                                                            -x_(2,1,0,0)x_(4,0,0,0)+x_(2,0,0,0)x_(4,1,0,0)                                               0                                                                                            -x_(3,1,0,0)x_(4,0,0,0)+x_(3,0,0,0)x_(4,1,0,0)                                               0                                                                                            |
     | -x_(0,1,0,1)x_(1,0,0,0)-x_(0,1,0,0)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,0)+x_(0,0,0,0)x_(1,1,0,1) -x_(0,1,0,0)x_(1,0,0,0)+x_(0,0,0,0)x_(1,1,0,0)                                               -x_(0,1,0,1)x_(2,0,0,0)-x_(0,1,0,0)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,0)+x_(0,0,0,0)x_(2,1,0,1) -x_(0,1,0,0)x_(2,0,0,0)+x_(0,0,0,0)x_(2,1,0,0)                                               -x_(1,1,0,1)x_(2,0,0,0)-x_(1,1,0,0)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,0)+x_(1,0,0,0)x_(2,1,0,1) -x_(1,1,0,0)x_(2,0,0,0)+x_(1,0,0,0)x_(2,1,0,0)                                               -x_(0,1,0,1)x_(3,0,0,0)-x_(0,1,0,0)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,0)+x_(0,0,0,0)x_(3,1,0,1) -x_(0,1,0,0)x_(3,0,0,0)+x_(0,0,0,0)x_(3,1,0,0)                                               -x_(1,1,0,1)x_(3,0,0,0)-x_(1,1,0,0)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,0)+x_(1,0,0,0)x_(3,1,0,1) -x_(1,1,0,0)x_(3,0,0,0)+x_(1,0,0,0)x_(3,1,0,0)                                               -x_(2,1,0,1)x_(3,0,0,0)-x_(2,1,0,0)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,0)+x_(2,0,0,0)x_(3,1,0,1) -x_(2,1,0,0)x_(3,0,0,0)+x_(2,0,0,0)x_(3,1,0,0)                                               -x_(0,1,0,1)x_(4,0,0,0)-x_(0,1,0,0)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,0)+x_(0,0,0,0)x_(4,1,0,1) -x_(0,1,0,0)x_(4,0,0,0)+x_(0,0,0,0)x_(4,1,0,0)                                               -x_(1,1,0,1)x_(4,0,0,0)-x_(1,1,0,0)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,0)+x_(1,0,0,0)x_(4,1,0,1) -x_(1,1,0,0)x_(4,0,0,0)+x_(1,0,0,0)x_(4,1,0,0)                                               -x_(2,1,0,1)x_(4,0,0,0)-x_(2,1,0,0)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,0)+x_(2,0,0,0)x_(4,1,0,1) -x_(2,1,0,0)x_(4,0,0,0)+x_(2,0,0,0)x_(4,1,0,0)                                               -x_(3,1,0,1)x_(4,0,0,0)-x_(3,1,0,0)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,0)+x_(3,0,0,0)x_(4,1,0,1) -x_(3,1,0,0)x_(4,0,0,0)+x_(3,0,0,0)x_(4,1,0,0)                                               |
     | -x_(0,1,0,1)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,1)                                               -x_(0,1,0,1)x_(1,0,0,0)-x_(0,1,0,0)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,0)+x_(0,0,0,0)x_(1,1,0,1) -x_(0,1,0,1)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,1)                                               -x_(0,1,0,1)x_(2,0,0,0)-x_(0,1,0,0)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,0)+x_(0,0,0,0)x_(2,1,0,1) -x_(1,1,0,1)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,1)                                               -x_(1,1,0,1)x_(2,0,0,0)-x_(1,1,0,0)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,0)+x_(1,0,0,0)x_(2,1,0,1) -x_(0,1,0,1)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,1)                                               -x_(0,1,0,1)x_(3,0,0,0)-x_(0,1,0,0)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,0)+x_(0,0,0,0)x_(3,1,0,1) -x_(1,1,0,1)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,1)                                               -x_(1,1,0,1)x_(3,0,0,0)-x_(1,1,0,0)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,0)+x_(1,0,0,0)x_(3,1,0,1) -x_(2,1,0,1)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,1)                                               -x_(2,1,0,1)x_(3,0,0,0)-x_(2,1,0,0)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,0)+x_(2,0,0,0)x_(3,1,0,1) -x_(0,1,0,1)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,1)                                               -x_(0,1,0,1)x_(4,0,0,0)-x_(0,1,0,0)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,0)+x_(0,0,0,0)x_(4,1,0,1) -x_(1,1,0,1)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,1)                                               -x_(1,1,0,1)x_(4,0,0,0)-x_(1,1,0,0)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,0)+x_(1,0,0,0)x_(4,1,0,1) -x_(2,1,0,1)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,1)                                               -x_(2,1,0,1)x_(4,0,0,0)-x_(2,1,0,0)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,0)+x_(2,0,0,0)x_(4,1,0,1) -x_(3,1,0,1)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,1)                                               -x_(3,1,0,1)x_(4,0,0,0)-x_(3,1,0,0)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,0)+x_(3,0,0,0)x_(4,1,0,1) |
     | 0                                                                                            -x_(0,1,0,1)x_(1,0,0,1)+x_(0,0,0,1)x_(1,1,0,1)                                               0                                                                                            -x_(0,1,0,1)x_(2,0,0,1)+x_(0,0,0,1)x_(2,1,0,1)                                               0                                                                                            -x_(1,1,0,1)x_(2,0,0,1)+x_(1,0,0,1)x_(2,1,0,1)                                               0                                                                                            -x_(0,1,0,1)x_(3,0,0,1)+x_(0,0,0,1)x_(3,1,0,1)                                               0                                                                                            -x_(1,1,0,1)x_(3,0,0,1)+x_(1,0,0,1)x_(3,1,0,1)                                               0                                                                                            -x_(2,1,0,1)x_(3,0,0,1)+x_(2,0,0,1)x_(3,1,0,1)                                               0                                                                                            -x_(0,1,0,1)x_(4,0,0,1)+x_(0,0,0,1)x_(4,1,0,1)                                               0                                                                                            -x_(1,1,0,1)x_(4,0,0,1)+x_(1,0,0,1)x_(4,1,0,1)                                               0                                                                                            -x_(2,1,0,1)x_(4,0,0,1)+x_(2,0,0,1)x_(4,1,0,1)                                               0                                                                                            -x_(3,1,0,1)x_(4,0,0,1)+x_(3,0,0,1)x_(4,1,0,1)                                               |

             4       20
o6 : Matrix S  <--- S

See also

Ways to use flattenedGenericTensor :